Reduced Length Least-Squares AR Parameter Estimation
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Solution Overview
Problem
Existing AR models require a large number of parameters to accurately model long correlations, leading to increased complexity and inefficiency, as they need to match autocorrelations up to a certain length, which is impractical for modeling stochastic processes with longer correlations.
Innovation Solution
The use of reduced length least-squares (LS) autoregressive (AR) parameter estimation, where a pth order AR process models stochastic processes with autocorrelations of length m, where p is significantly less than m, allowing for efficient characterization of long correlations using a smaller number of parameters.
Engineering Contradictions & Design Principles
Engineering Contradiction Analysis
1Measurement precision
If a traditional AR model is used to model long correlations, then the autocorrelation matching accuracy is improved, but the model complexity increases
Solution Approach 1:
The patent changes the parameter estimation method from Yule-Walker to least-squares, and modifies the objective function to minimize the difference between modeled and actual autocorrelations. This parameter change allows achieving better autocorrelation matching with reduced model order, thereby reducing complexity while maintaining or improving accuracy
Solution Approach 2:
The patent introduces a dynamic model order selection mechanism where the optimal AR model order is determined based on the correlation length of the stochastic process. This dynamic adaptation allows the model to use lower complexity for shorter correlations and higher complexity only when necessary for longer correlations
2Measurement precision
If a large AR model order p is used to model long correlations, then the autocorrelation matching capability is improved, but the computational efficiency deteriorates
Solution Approach 1:
The patent changes the parameter estimation approach to least-squares method with a specifically designed objective function that emphasizes matching longer autocorrelations. This parameter change enables achieving good autocorrelation matching with smaller model order p, thus improving computational efficiency while maintaining matching capability
Solution Approach 2:
The patent applies partial action by selecting only the essential autocorrelation lags that contribute most to the stochastic process characteristics. Instead of matching all autocorrelations equally, the method focuses on the most significant ones, reducing the computational burden while maintaining effective modeling
3Measurement precision
If the AR model order p is increased to match autocorrelations up to length p, then the correlation modeling accuracy is improved, but the practical implementability deteriorates
Solution Approach 1:
The patent changes the optimization criterion to a least-squares objective function that can be solved efficiently using standard algorithms. This parameter change makes the solution computationally tractable and easier to implement in practical applications compared to traditional methods that require solving complex nonlinear equations
Solution Approach 2:
The patent uses a simplified AR model structure that copies only the essential characteristics of the stochastic process rather than attempting to exactly reproduce all correlations. This copying approach maintains practical implementability while achieving sufficient modeling accuracy for most applications
Data Source
AI summary
An apparatus and method for modelling a random process using reduced length least-squares autoregressive parameter estimation is herein disclosed. The apparatus includes an autocorrelation processor, configured to generate or estimate autocorrelations of length m for a stochastic process, where m is an integer; and a least-squares (LS) estimation processor connected to the autocorrelation processor and configured to model the stochastic process by estimating pth order autoregressive (AR) parameters using LS regression, where p is an integer much less than m. The method includes generating, by an autocorrelation processor, autocorrelations of length m for a stochastic process, where m is an integer; and modelling the stochastic process, by a least-squares estimation processor, by estimating pth order autoregressive (AR) parameters by least-squares (LS) regression, where p is an integer much less than m.